| L(s) = 1 | − 3·2-s − 9·3-s + 4-s − 2·5-s + 27·6-s + 6·8-s + 34·9-s + 6·10-s + 26·11-s − 9·12-s + 30·13-s + 18·15-s − 16-s − 42·17-s − 102·18-s + 25·19-s − 2·20-s − 78·22-s + 8·23-s − 54·24-s + 31·25-s − 90·26-s − 63·27-s − 12·29-s − 54·30-s − 27·32-s − 234·33-s + ⋯ |
| L(s) = 1 | − 3/2·2-s − 3·3-s + 1/4·4-s − 2/5·5-s + 9/2·6-s + 3/4·8-s + 34/9·9-s + 3/5·10-s + 2.36·11-s − 3/4·12-s + 2.30·13-s + 6/5·15-s − 0.0625·16-s − 2.47·17-s − 5.66·18-s + 1.31·19-s − 0.0999·20-s − 3.54·22-s + 8/23·23-s − 9/4·24-s + 1.23·25-s − 3.46·26-s − 7/3·27-s − 0.413·29-s − 9/5·30-s − 0.843·32-s − 7.09·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47045881 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47045881 ^{s/2} \, \Gamma_{\C}(s+1)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.07446304393\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07446304393\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 - 25 T + 54 p T^{2} - 47 p^{2} T^{3} + 54 p^{3} T^{4} - 25 p^{4} T^{5} + p^{6} T^{6} \) |
| good | 2 | \( 1 + 3 T + p^{3} T^{2} + 15 T^{3} + 5 p^{2} T^{4} + 27 T^{5} - T^{6} + 27 p^{2} T^{7} + 5 p^{6} T^{8} + 15 p^{6} T^{9} + p^{11} T^{10} + 3 p^{10} T^{11} + p^{12} T^{12} \) |
| 3 | \( 1 + p^{2} T + 47 T^{2} + 20 p^{2} T^{3} + 463 T^{4} + 11 p^{4} T^{5} + 238 p^{2} T^{6} + 11 p^{6} T^{7} + 463 p^{4} T^{8} + 20 p^{8} T^{9} + 47 p^{8} T^{10} + p^{12} T^{11} + p^{12} T^{12} \) |
| 5 | \( 1 + 2 T - 27 T^{2} + 114 T^{3} + 238 T^{4} - 2656 T^{5} + 5401 T^{6} - 2656 p^{2} T^{7} + 238 p^{4} T^{8} + 114 p^{6} T^{9} - 27 p^{8} T^{10} + 2 p^{10} T^{11} + p^{12} T^{12} \) |
| 7 | \( ( 1 + 69 T^{2} - 94 T^{3} + 69 p^{2} T^{4} + p^{6} T^{6} )^{2} \) |
| 11 | \( ( 1 - 13 T + 280 T^{2} - 3149 T^{3} + 280 p^{2} T^{4} - 13 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 13 | \( 1 - 30 T + 779 T^{2} - 14370 T^{3} + 239570 T^{4} - 3505902 T^{5} + 48205499 T^{6} - 3505902 p^{2} T^{7} + 239570 p^{4} T^{8} - 14370 p^{6} T^{9} + 779 p^{8} T^{10} - 30 p^{10} T^{11} + p^{12} T^{12} \) |
| 17 | \( 1 + 42 T + 333 T^{2} + 6726 T^{3} + 513306 T^{4} + 6959454 T^{5} + 21390509 T^{6} + 6959454 p^{2} T^{7} + 513306 p^{4} T^{8} + 6726 p^{6} T^{9} + 333 p^{8} T^{10} + 42 p^{10} T^{11} + p^{12} T^{12} \) |
| 23 | \( 1 - 8 T - 1413 T^{2} + 3876 T^{3} + 1337428 T^{4} - 1325942 T^{5} - 814307171 T^{6} - 1325942 p^{2} T^{7} + 1337428 p^{4} T^{8} + 3876 p^{6} T^{9} - 1413 p^{8} T^{10} - 8 p^{10} T^{11} + p^{12} T^{12} \) |
| 29 | \( 1 + 12 T + 1091 T^{2} + 12516 T^{3} + 342470 T^{4} + 25875306 T^{5} + 53097851 T^{6} + 25875306 p^{2} T^{7} + 342470 p^{4} T^{8} + 12516 p^{6} T^{9} + 1091 p^{8} T^{10} + 12 p^{10} T^{11} + p^{12} T^{12} \) |
| 31 | \( 1 - 1222 T^{2} + 2797835 T^{4} - 2253009856 T^{6} + 2797835 p^{4} T^{8} - 1222 p^{8} T^{10} + p^{12} T^{12} \) |
| 37 | \( 1 - 5190 T^{2} + 13520751 T^{4} - 22697438888 T^{6} + 13520751 p^{4} T^{8} - 5190 p^{8} T^{10} + p^{12} T^{12} \) |
| 41 | \( 1 - 63 T + 4739 T^{2} - 215208 T^{3} + 233233 p T^{4} - 413399385 T^{5} + 16368671414 T^{6} - 413399385 p^{2} T^{7} + 233233 p^{5} T^{8} - 215208 p^{6} T^{9} + 4739 p^{8} T^{10} - 63 p^{10} T^{11} + p^{12} T^{12} \) |
| 43 | \( 1 + 34 T - 3475 T^{2} - 31338 T^{3} + 9673978 T^{4} - 28877782 T^{5} - 22052754731 T^{6} - 28877782 p^{2} T^{7} + 9673978 p^{4} T^{8} - 31338 p^{6} T^{9} - 3475 p^{8} T^{10} + 34 p^{10} T^{11} + p^{12} T^{12} \) |
| 47 | \( 1 - 58 T - 3429 T^{2} + 99066 T^{3} + 16840372 T^{4} - 217034296 T^{5} - 36013746767 T^{6} - 217034296 p^{2} T^{7} + 16840372 p^{4} T^{8} + 99066 p^{6} T^{9} - 3429 p^{8} T^{10} - 58 p^{10} T^{11} + p^{12} T^{12} \) |
| 53 | \( 1 + 12 T + 7659 T^{2} + 91332 T^{3} + 36866070 T^{4} + 510480060 T^{5} + 120117765391 T^{6} + 510480060 p^{2} T^{7} + 36866070 p^{4} T^{8} + 91332 p^{6} T^{9} + 7659 p^{8} T^{10} + 12 p^{10} T^{11} + p^{12} T^{12} \) |
| 59 | \( 1 + 147 T + 16901 T^{2} + 1425606 T^{3} + 106851335 T^{4} + 7433603271 T^{5} + 456697800146 T^{6} + 7433603271 p^{2} T^{7} + 106851335 p^{4} T^{8} + 1425606 p^{6} T^{9} + 16901 p^{8} T^{10} + 147 p^{10} T^{11} + p^{12} T^{12} \) |
| 61 | \( 1 - 58 T - 5011 T^{2} + 428958 T^{3} + 12450718 T^{4} - 1018447028 T^{5} + 4642755289 T^{6} - 1018447028 p^{2} T^{7} + 12450718 p^{4} T^{8} + 428958 p^{6} T^{9} - 5011 p^{8} T^{10} - 58 p^{10} T^{11} + p^{12} T^{12} \) |
| 67 | \( 1 - 3 p T + 28649 T^{2} - 45546 p T^{3} + 282765251 T^{4} - 23237371929 T^{5} + 1659244019666 T^{6} - 23237371929 p^{2} T^{7} + 282765251 p^{4} T^{8} - 45546 p^{7} T^{9} + 28649 p^{8} T^{10} - 3 p^{11} T^{11} + p^{12} T^{12} \) |
| 71 | \( 1 + 102 T + 19395 T^{2} + 1624554 T^{3} + 208411794 T^{4} + 13817724054 T^{5} + 1288263668227 T^{6} + 13817724054 p^{2} T^{7} + 208411794 p^{4} T^{8} + 1624554 p^{6} T^{9} + 19395 p^{8} T^{10} + 102 p^{10} T^{11} + p^{12} T^{12} \) |
| 73 | \( 1 - 7 T - 12713 T^{2} + 22960 T^{3} + 94466917 T^{4} + 36786071 T^{5} - 563497579498 T^{6} + 36786071 p^{2} T^{7} + 94466917 p^{4} T^{8} + 22960 p^{6} T^{9} - 12713 p^{8} T^{10} - 7 p^{10} T^{11} + p^{12} T^{12} \) |
| 79 | \( 1 + 12035 T^{2} + 69730790 T^{4} + 4384013520 T^{5} + 462010537511 T^{6} + 4384013520 p^{2} T^{7} + 69730790 p^{4} T^{8} + 12035 p^{8} T^{10} + p^{12} T^{12} \) |
| 83 | \( ( 1 - 73 T + 15082 T^{2} - 608183 T^{3} + 15082 p^{2} T^{4} - 73 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 89 | \( 1 + 72 T + 9723 T^{2} + 575640 T^{3} + 40073838 T^{4} + 8635069188 T^{5} + 306038573143 T^{6} + 8635069188 p^{2} T^{7} + 40073838 p^{4} T^{8} + 575640 p^{6} T^{9} + 9723 p^{8} T^{10} + 72 p^{10} T^{11} + p^{12} T^{12} \) |
| 97 | \( 1 - 21 T + 12695 T^{2} - 263508 T^{3} + 43921757 T^{4} - 6267660039 T^{5} - 57670086154 T^{6} - 6267660039 p^{2} T^{7} + 43921757 p^{4} T^{8} - 263508 p^{6} T^{9} + 12695 p^{8} T^{10} - 21 p^{10} T^{11} + p^{12} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.95734516537837958099481812246, −10.82823240938914621247752772732, −10.71476085646487899152510447752, −10.43493749144900231436596002593, −9.668510674184599677563624161801, −9.549447942268131998036406479746, −9.378271700993899991111338823306, −9.205692631029819668959370032924, −8.921182567136291675618346232195, −8.508360898295052111166777958608, −8.317286874476701056503047280101, −8.167788986407499217854790031268, −7.52892446315695115894033998545, −6.99085913654587462377482131703, −6.93697368223925667823325338341, −6.44748239747141298943803001821, −6.33304559245663433956941880392, −5.92418137942803284415178909158, −5.90275402424136128608612692557, −5.15779616186179174069314923219, −4.77357460935657086079466821249, −4.57749720574230799147329532250, −3.68820208513988294988307004110, −3.65326289968477166989528342053, −1.13742909390690919982331790996,
1.13742909390690919982331790996, 3.65326289968477166989528342053, 3.68820208513988294988307004110, 4.57749720574230799147329532250, 4.77357460935657086079466821249, 5.15779616186179174069314923219, 5.90275402424136128608612692557, 5.92418137942803284415178909158, 6.33304559245663433956941880392, 6.44748239747141298943803001821, 6.93697368223925667823325338341, 6.99085913654587462377482131703, 7.52892446315695115894033998545, 8.167788986407499217854790031268, 8.317286874476701056503047280101, 8.508360898295052111166777958608, 8.921182567136291675618346232195, 9.205692631029819668959370032924, 9.378271700993899991111338823306, 9.549447942268131998036406479746, 9.668510674184599677563624161801, 10.43493749144900231436596002593, 10.71476085646487899152510447752, 10.82823240938914621247752772732, 10.95734516537837958099481812246
Plot not available for L-functions of degree greater than 10.
